[Paper Review] Quantum Search Algorithm for Binary Constant Weight Codes

Kein Yukiyoshi, Naoki Ishikawa|arXiv (Cornell University)|Nov 9, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a novel quantum search algorithm for binary constant weight codes using Quadratic Unconstrained Binary Optimization (QUBO) and Grover Adaptive Search (GAS), achieving a 63% reduction in classical query complexity and 31% in quantum query complexity. The method reduces qubit requirements via algebraic structure exploitation and establishes tight bounds on solution counts and objective function minima.

ABSTRACT

A binary constant weight code is a type of error-correcting code with a wide range of applications. The problem of finding a binary constant weight code has long been studied as a combinatorial optimization problem in coding theory. In this paper, we propose a quantum search algorithm for binary constant weight codes. Specifically, the search problem is newly formulated as a quadratic unconstrained binary optimization (QUBO) and Grover adaptive search (GAS) is used for providing the quadratic speedup. Focusing on the inherent structure of the problem, we derive an upper bound on the minimum of the objective function value and a lower bound on the exact number of solutions. In our algebraic analysis, it was found that this proposed algorithm is capable of reducing the number of required qubits, thus enhancing the feasibility. Additionally, our simulations demonstrated that it reduces the query complexities by 63% in the classical domain and 31% in the quantum domain. The proposed approach may be useful for other quantum search algorithms and optimization problems.

Motivation & Objective

• To address the combinatorial explosion in finding optimal binary constant weight codes, a major challenge in coding theory.
• To formulate the constant weight code problem as a QUBO problem to enable quantum optimization.
• To reduce the number of required qubits in quantum circuit implementation through structural analysis.
• To derive tight upper and lower bounds on the objective function and solution count to enhance algorithm efficiency.
• To achieve quadratic speedup in search for optimal codes using Grover adaptive search (GAS).

Proposed method

• The constant weight code problem is reformulated as a QUBO problem using inner products to represent Hamming distances, replacing XOR-based representations.
• A novel objective function is constructed to enforce a minimum distance $ d $ between codewords via inner product constraints.
• Grover Adaptive Search (GAS) is applied to the QUBO formulation to achieve quadratic speedup in finding optimal codes.
• Algebraic analysis is used to derive an upper bound on the minimum objective function value and a lower bound on the exact number of solutions.
• The method exploits structural constraints—such as column weight and codeword distribution—to minimize qubit requirements.
• Simulations validate the performance gains in both classical and quantum query complexity under the proposed formulation.

Experimental results

Research questions

• RQ1 Can the binary constant weight code construction problem be effectively reformulated as a QUBO problem for quantum optimization?
• RQ2 How can the inherent algebraic structure of constant weight codes be leveraged to reduce qubit count and improve feasibility?
• RQ3 What are tight bounds on the minimum objective function value and the exact number of valid solutions for a given code?
• RQ4 To what extent does the proposed GAS-based algorithm reduce query complexity compared to classical and standard quantum approaches?
• RQ5 Can the proposed method be generalized to other combinatorial optimization problems in coding theory?

Key findings

• The proposed QUBO formulation reduces the number of required qubits by exploiting structural constraints of constant weight codes.
• The algorithm achieves a 63% reduction in classical query complexity and a 31% reduction in quantum query complexity compared to baseline methods.
• An upper bound on the minimum objective function value is derived, enabling tighter optimization bounds.
• A lower bound on the exact number of solutions is established as $ oxed{\underline{t} = \min_{2 \leq i \leq w - d/2} \binom{w}{i}} $ when $ w - d/2 \geq 2 $, and $ w! $ when $ w - d/2 = 1 $.
• Theoretical analysis confirms that the proposed GAS variant accelerates quadratic speedup by leveraging problem-specific structure.
• Simulations demonstrate consistent performance gains across multiple parameter sets, validating the theoretical bounds and efficiency improvements.